The Riemann-Roch theorem for elliptic operators and solvability of elliptic equations with additional conditions on compact subsets
The Robbins problem for elliptic equations in space
The Role of Green’s Functions in Inverse Scattering at Fixed Energy
The role of the boundary in some semilinear Neumann problems
The Scattering of Classical Waves from Inhomogeneous media.
The Schrödinger equation on a compact manifold : Strichartz estimates and applications
We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any riemannian compact manifold. As a consequence we infer global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three-manifolds in the case of quadratic nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres.
The Schrödinger–Maxwell system with Dirac mass
The simple layer potential for the biharmonic equation in variables
A theory of the «simple layer potential» for the classical biharmonic problem in is worked out. This hinges on the study of a new class of singular integral operators, each of them trasforming a vector with scalar components into a vector whose components are differential forms of degree one.
The Sinc Method in Multiple Space Dimensions: Model Probelms.
The solution of Kato's conjecture (after Auscher, Hofmann, Lacey, McIntosh and Tchamitchian)
Kato’s conjecture, stating that the domain of the square root of any accretive operator with bounded measurable coefficients in is the Sobolev space , i.e. the domain of the underlying sesquilinear form, has recently been obtained by Auscher, Hofmann, Lacey, McIntosh and the author. These notes present the result and explain the strategy of proof.
The solution of the Kato problem in two dimensions.
We solve, in two dimensions, the "square root problem of Kato". That is, for L ≡ -div (A(x)∇), where A(x) is a 2 x 2 accretive matrix of bounded measurable complex coefficients, we prove that L1/2: L12(R2) → L2(R2).[Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].
The solution of the third problem for the Laplace equation on planar domains with smooth boundary and inside cracks and modified jump conditions on cracks.
The spaces of quantum states for some harmonic oscillator potentials on the punctured plane C∖{0}
We consider equivariant solutions of Schrödinger equations on C∖{0} with harmonic oscillator potentials. We determine the spaces of equivariant quantum states in three cases: for an isotropic and anisotropic harmonic oscillator potential centered at 0, and for a potential not centered at 0.
The spectral cutting problem
The spectral distribution of a globally elliptic operator
The spectral properties of the Schrödinger operator in nonseparable Hilbert spaces
The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential.
The spectrum of Schrödinger operators with random magnetic fields
We shall consider the Schrödinger operators on with the magnetic field given by a nonnegative constant field plus random magnetic fields of the Anderson type or of the Poisson-Anderson type. We shall investigate the spectrum of these operators by the method of the admissible potentials by Kirsch-Martinelli. Moreover, we shall prove the lower Landau levels are infinitely degenerated eigenvalues when the constant field is sufficiently large, by estimating the growth order of the eigenfunctions...
The Spectrum of the Continuous Laplacian on a Graph.
The speed of propagation for KPP type problems. I: Periodic framework
This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diffusion equations with Kolmogorov–Petrovsky–Piskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a follow-up of the article [7]. It is proved that the minimal speed of pulsating fronts is given by a variational formula involving linear eigenvalue problems. Some consequences concerning the influence of the geometry of the domain,...